Use logic, not guess and check, to find where the numbers from 1 to 12 belong in both the first column and the top row so that the puzzle acts like a multiplication table. Can you do it, or will some of the clues trick you?

Print the puzzles or type the solution in this excel file: 12 factors 1251-1258

Now I’ll share some facts about the number 1256:

• 1256 is a composite number.
• Prime factorization: 1256 = 2 × 2 × 2 × 157, which can be written 1256 = 2³ × 157
• The exponents in the prime factorization are 3 and 1. Adding one to each and multiplying we get (3 + 1)(1 + 1) = 4 × 2 = 8. Therefore 1256 has exactly 8 factors.
• Factors of 1256: 1, 2, 4, 8, 157, 314, 628, 1256
• Factor pairs: 1256 = 1 × 1256, 2 × 628, 4 × 314, or 8 × 157
• Taking the factor pair with the largest square number factor, we get √1256 = (√4)(√314) = 2√314 ≈ 35.44009

1256 is the sum of two squares:
34² + 10² = 1256

1256 is the hypotenuse of a Pythagorean triple:
680-1056-1256 which is 8 times (85-132-157) and
can also be calculated from 2(34)(10), 34² – 10², 34² + 10²

1256 is 888 in BASE 12 because 8(144 + 12 + 1) = 8(157) = 1256

Find the common factor of 8 and 80 so that only numbers from 1 to 12 will be put in the top row of this multiplication table puzzle. Then work down row by row writing the factors of each clue so that the numbers from 1 to 12 appear only once in both the first column and the top row. You can do this!

Print the puzzles or type the solution in this excel file: 12 factors 1251-1258

Here are a few facts about the post number, 1254:

• 1254 is a composite number.
• Prime factorization: 1254 = 2 × 3 × 11 × 19
• The exponents in the prime factorization are 1, 1, 1, and 1. Adding one to each and multiplying we get (1 + 1)(1 + 1)(1 + 1)(1 + 1) = 2 × 2 × 2 × 2 = 16. Therefore 1254 has exactly 16 factors.
• Factors of 1254: 1, 2, 3, 6, 11, 19, 22, 33, 38, 57, 66, 114, 209, 418, 627, 1254
• Factor pairs: 1254 = 1 × 1254, 2 × 627, 3 × 418, 6 × 209, 11 × 114, 19 × 66, 22 × 57, or 33 × 38
• 1254 has no square factors that allow its square root to be simplified. √1254 ≈ 35.41186

1254 is the sum of the twenty-four prime numbers from 7 to 103. Do you know what those prime numbers are?

In what order should the numbers from 1 to 12 be written in the first column and also in the top row so that this puzzle works like a multiplication table?

Print the puzzles or type the solution in this excel file: 12 factors 1251-1258

Now I’ll tell you a little bit about the number 1253:

• 1253 is a composite number.
• Prime factorization: 1253 = 7 × 179
• The exponents in the prime factorization are 1 and 1. Adding one to each and multiplying we get (1 + 1)(1 + 1) = 2 × 2 = 4. Therefore 1253 has exactly 4 factors.
• Factors of 1253: 1, 7, 179, 1253
• Factor pairs: 1253 = 1 × 1253 or 7 × 179
• 1253 has no square factors that allow its square root to be simplified. √1253 ≈ 35.39774

1253 is also the sum of the eleven prime numbers from 89 to 139. Do you know what those prime numbers are?

Other than 1, what is the common factor of all the clues in this puzzle? Use that answer to fill in all the cells in the first column and the top row with the numbers from 1 to 12. then you will have the start of a different kind of multiplication table.

Print the puzzles or type the solution in this excel file: 12 factors 1251-1258

Now I’ll share some facts about the number 1251:

• 1251 is a composite number.
• Prime factorization: 1251 = 3 × 3 × 139, which can be written 1251 = 3² × 139
• The exponents in the prime factorization are 2 and 1. Adding one to each and multiplying we get (2 + 1)(1 + 1) = 3 × 2  = 6. Therefore 1251 has exactly 6 factors.
• Factors of 1251: 1, 3, 9, 139, 417, 1251
• Factor pairs: 1251 = 1 × 1251, 3 × 417, or 9 × 139
• Taking the factor pair with the largest square number factor, we get √1251 = (√9)(√139) = 3√139 ≈ 35.36948

1251 is also the sum of five consecutive prime numbers:
239 + 241 + 251 + 257 + 263 = 1251

The clues in one of the columns for this puzzle as well as one of the rows are 9 and 3. You will need to figure out where to put the factors 1, 3, 3, and 9 to make those clues work. You might think it doesn’t matter where you write those factors, but believe me, it does matter. My advice: Don’t start with those clues. Find another logical place to start. Good luck with this one!

Print the puzzles or type the solution in this excel file: 10-factors-1242-1250

Here are some facts about the number 1250:

• 1250 is a composite number.
• Prime factorization: 1250 = 2 × 5 × 5 × 5 × 5, which can be written 1250 = 2 × 5⁴
• The exponents in the prime factorization are 1 and 5. Adding one to each and multiplying we get (1 + 1)(4 + 1) = 2 × 5 = 10. Therefore 1250 has exactly 10 factors.
• Factors of 1250: 1, 2, 5, 10, 25, 50, 125, 250, 625, 1250
• Factor pairs: 1250 = 1 × 1250, 2 × 625, 5 × 250, 10 × 125, or 25 × 50
• Taking the factor pair with the largest square number factor, we get √1250 = (√625)(√2) = 25√2 ≈ 35.35534

1250 is the sum of consecutive squares two different ways:
193 + 197 + 199 + 211 + 223 + 227 = 1250
619 + 631 = 1250

1250 is the sum of two squares THREE different ways:
31² + 17² = 1250
25² + 25² = 1250
35² + 5² = 1250

1250 is the hypotenuse of FOUR Pythagorean triples:
750-1000-1250 which is (3-4-5) times 250,
672-1054-1250 which is 2 times (336-527-625) and
can also be calculated from 31² – 17², 2(31)(17), 31² + 17²,
440-1170-1250 which is 10 times (44-117-125), and
350-1200-1250 which is (7-24-25) times 50 and
can also be calculated from 2(35)(5), 35² – 5², 35² + 5²